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Properties

Label 245.6.b.g
Level $245$
Weight $6$
Character orbit 245.b
Analytic conductor $39.294$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,6,Mod(99,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 576 q^{4} - 3460 q^{9} + 1556 q^{11} + 1196 q^{15} + 8824 q^{16} - 7868 q^{25} - 27276 q^{29} - 17072 q^{30} + 174040 q^{36} - 58036 q^{39} - 96440 q^{44} + 155560 q^{46} - 170304 q^{50} + 94852 q^{51}+ \cdots - 882336 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 − 10.4629i − 28.3615i −77.4731 −5.99023 + 55.5798i −296.745 0 475.782i −561.377 581.528 + 62.6754i
99.2 − 10.4629i 28.3615i −77.4731 5.99023 − 55.5798i 296.745 0 475.782i −561.377 −581.528 − 62.6754i
99.3 − 10.2318i − 7.10531i −72.6906 −40.4579 + 38.5766i −72.7004 0 416.339i 192.515 394.710 + 413.959i
99.4 − 10.2318i 7.10531i −72.6906 40.4579 − 38.5766i 72.7004 0 416.339i 192.515 −394.710 − 413.959i
99.5 − 8.36549i − 29.5396i −37.9815 43.8645 − 34.6541i −247.113 0 50.0380i −629.586 −289.898 − 366.948i
99.6 − 8.36549i 29.5396i −37.9815 −43.8645 + 34.6541i 247.113 0 50.0380i −629.586 289.898 + 366.948i
99.7 − 7.24652i − 3.38772i −20.5121 −50.5162 − 23.9397i −24.5492 0 − 83.2477i 231.523 −173.480 + 366.067i
99.8 − 7.24652i 3.38772i −20.5121 50.5162 + 23.9397i 24.5492 0 − 83.2477i 231.523 173.480 − 366.067i
99.9 − 5.97543i − 0.602866i −3.70572 −22.0505 − 51.3690i −3.60238 0 − 169.070i 242.637 −306.952 + 131.761i
99.10 − 5.97543i 0.602866i −3.70572 22.0505 + 51.3690i 3.60238 0 − 169.070i 242.637 306.952 − 131.761i
99.11 − 4.76692i − 22.8362i 9.27648 12.4215 − 54.5042i −108.858 0 − 196.762i −278.491 −259.817 − 59.2122i
99.12 − 4.76692i 22.8362i 9.27648 −12.4215 + 54.5042i 108.858 0 − 196.762i −278.491 259.817 + 59.2122i
99.13 − 2.19823i − 17.6013i 27.1678 −53.6239 + 15.7949i −38.6917 0 − 130.064i −66.8056 34.7209 + 117.878i
99.14 − 2.19823i 17.6013i 27.1678 53.6239 − 15.7949i 38.6917 0 − 130.064i −66.8056 −34.7209 − 117.878i
99.15 − 0.285313i − 15.4407i 31.9186 43.0320 + 35.6826i −4.40543 0 − 18.2368i 4.58487 10.1807 − 12.2776i
99.16 − 0.285313i 15.4407i 31.9186 −43.0320 − 35.6826i 4.40543 0 − 18.2368i 4.58487 −10.1807 + 12.2776i
99.17 0.285313i − 15.4407i 31.9186 −43.0320 + 35.6826i 4.40543 0 18.2368i 4.58487 −10.1807 − 12.2776i
99.18 0.285313i 15.4407i 31.9186 43.0320 − 35.6826i −4.40543 0 18.2368i 4.58487 10.1807 + 12.2776i
99.19 2.19823i − 17.6013i 27.1678 53.6239 + 15.7949i 38.6917 0 130.064i −66.8056 −34.7209 + 117.878i
99.20 2.19823i 17.6013i 27.1678 −53.6239 − 15.7949i −38.6917 0 130.064i −66.8056 34.7209 − 117.878i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.6.b.g ✓ 32
5.b even 2 1 inner 245.6.b.g ✓ 32
7.b odd 2 1 inner 245.6.b.g ✓ 32
35.c odd 2 1 inner 245.6.b.g ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.6.b.g ✓ 32 1.a even 1 1 trivial
245.6.b.g ✓ 32 5.b even 2 1 inner
245.6.b.g ✓ 32 7.b odd 2 1 inner
245.6.b.g ✓ 32 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{16} + 400 T_{2}^{14} + 63793 T_{2}^{12} + 5185294 T_{2}^{10} + 227703580 T_{2}^{8} + \cdots + 13441849344 \) Copy content Toggle raw display
\( T_{19}^{16} - 24559120 T_{19}^{14} + 225629684070592 T_{19}^{12} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display